These semiregular polyhedra of the second kind are the duals of the Archimedes' polyhedra. Their faces are superimposable (but not regular) and tangent to a sphere (inscribed sphere); their vertices are regular (of two or three types).
Each drawing provides a link to a popup applet (more about the characteristics of the semiregular polyhedra).

The dual of the cuboctahedron (resp. icosidodecahedron) has 12 (resp. 30) rhombic faces.
Its vertices of order 3 are the vertices of a cube (resp. a regular dodecahedron); the remaining vertices are those of a regular octahedron (resp. icosahedron). Thus these polyhedra can be made by assembling regular pyramids with a square (resp. pentagonal) base on each face of a cube (resp. a regular dodecahedron), or by assembling tetrahedra on the faces of a regular octahedron (resp. icosahedron).

 

The duals of the snub cube and of the snub dodecahedron have respectively 24 and 60 pentagonal faces.


These duals of the prisms and of the antiprisms complete the list of the second kind semiregular polyhedra.
(The diamond of order 4 is the regular octahedron; the antidiamond of order 3 is the cube.)
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